3) Other methods that can relax the stiffness (Super-time stepping or wave methods)

Super time stepping (STS; Alexiades et al. 1996) is a technique that can be used to accelerate explicit parabolic calculations. It is based on the stability properties of the Chebyshev polynomials, which allow us to relax the CFL condition and take a larger timestep, $\Delta t_{STS}$. The method uses two input parameters, n, a positive integer, and $\nu \in (0, 1)$, which is a damping factor. The timestep allowed by the STS method is then given by

$$\Delta t_{STS} = \frac{n}{2\sqrt{\nu}}\left[\frac{(1+\sqrt{\nu})^{2n}-(1-\sqrt{\nu})^{2n}}{(1+\sqrt{\nu})^{2n}+(1-\sqrt{\nu})^{2n}}\right] \Delta t_{CFL}$$

and is divided into $n$ sub-timesteps $\tau_i$ as

$$\Delta t_{STS} = \sum_{i=1}^n (\tau_i)$$

with

$$\tau_i = \Delta t_{CFL}\left[(\nu -1)\cos \left(\frac{\pi (2,i - 1)}{2,n} \right) + \nu + 1 \right]^{-1}$$

where  $\Delta t_{CFL}$ is the timestep of the parabolic problem given by the classic CFL condition. It is important to note that the intermediate values computed along the $n$ sub-timesteps have no approximating properties: it is only after the whole $\Delta t_{STS}$ has been reached that the results approximate the solution and, consequently, have a physical meaning.

The above expressions can be easily implemented to improve the calculation efficiency of parabolic terms; nevertheless, the drawback is that the STS is a first-order scheme in time. In addition, the method has two free parameters, $n$, and it is necessary to carefully choose their values to optimize the performance while keeping numerical stability. The maximum ratio $\Delta t_{STS}/\Delta t_{CFL}$ that can be reached for any given $n$ corresponds to the following limit:

{\lim}_{\nu \rightarrow 0} \frac{\Delta t_{STS}}{\Delta t_{CFL}} = {\lim}_{\nu \rightarrow 0} \frac{n}{2\sqrt{\nu}} \left[\frac{(1+\sqrt{\nu})^{2n}-(1-\sqrt{\nu})^{2n}}{(1+\sqrt{\nu})^{2n}+(1-\sqrt{\nu})^{2n}}\right] = n^2

In this limit, the CFL criterion would need $n^2$ steps to reach one $\Delta t_{STS}$, while the STS method requires just $n$, as explained above. Assuming a similar computing load for each step, whether, in the STS or the CFL-limited calculation, this implies that the STS method would be $n$ times faster than the simple CFL-limited one. However, it is necessary to impose a lower threshold for $\nu$ because  $\nu = 0$ is a stability limit. Choosing low values of $\nu$ can make the STS method very sensitive to round-off errors (Alexiades et al. 1996).